Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 99.4%

Time: 21.7s

Alternatives: 8

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U) (if (<= t_1 1e+303) t_1 U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1e+303) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 1e+303) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 1e+303:
		tmp = t_1
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 1e+303)
		tmp = t_1;
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+303)
		tmp = t_1;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 1e+303], t$95$1, U]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < -inf.0

   1. Initial program 5.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0 44.4%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 44.4%

         \[\leadsto \color{blue}{-U} \]

   4. Simplified 44.4%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 1e303

   1. Initial program 99.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   if 1e303 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 5.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf 46.5%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\right)\right)\\ \mathbf{if}\;J \leq -6.9 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-234}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* -2.0 (* t_0 (* J (hypot 1.0 (/ (/ U (* J 2.0)) t_0)))))))
   (if (<= J -6.9e-161)
     t_1
     (if (<= J -2e-310) U (if (<= J 6.2e-234) (- U) t_1)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * 2.0)) / t_0))));
	double tmp;
	if (J <= -6.9e-161) {
		tmp = t_1;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if (J <= 6.2e-234) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = -2.0 * (t_0 * (J * Math.hypot(1.0, ((U / (J * 2.0)) / t_0))));
	double tmp;
	if (J <= -6.9e-161) {
		tmp = t_1;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if (J <= 6.2e-234) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = -2.0 * (t_0 * (J * math.hypot(1.0, ((U / (J * 2.0)) / t_0))))
	tmp = 0
	if J <= -6.9e-161:
		tmp = t_1
	elif J <= -2e-310:
		tmp = U
	elif J <= 6.2e-234:
		tmp = -U
	else:
		tmp = t_1
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0)))))
	tmp = 0.0
	if (J <= -6.9e-161)
		tmp = t_1;
	elseif (J <= -2e-310)
		tmp = U;
	elseif (J <= 6.2e-234)
		tmp = Float64(-U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * 2.0)) / t_0))));
	tmp = 0.0;
	if (J <= -6.9e-161)
		tmp = t_1;
	elseif (J <= -2e-310)
		tmp = U;
	elseif (J <= 6.2e-234)
		tmp = -U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(t$95$0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -6.9e-161], t$95$1, If[LessEqual[J, -2e-310], U, If[LessEqual[J, 6.2e-234], (-U), t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -6.90000000000000002e-161 or 6.2000000000000003e-234 < J

   1. Initial program 81.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)} \]

   if -6.90000000000000002e-161 < J < -1.999999999999994e-310

   1. Initial program 26.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf 46.1%

      \[\leadsto \color{blue}{U} \]

   if -1.999999999999994e-310 < J < 6.2000000000000003e-234

   1. Initial program 19.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0 38.4%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 38.4%

         \[\leadsto \color{blue}{-U} \]

   4. Simplified 38.4%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -6.9 \cdot 10^{-161}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-234}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \end{array} \]

Alternative 3: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -1 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -7 \cdot 10^{-308}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (* U (/ 0.5 J)))))))
   (if (<= J -1e-156)
     t_0
     (if (<= J -7e-308) U (if (<= J 1.05e-49) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, (U * (0.5 / J))));
	double tmp;
	if (J <= -1e-156) {
		tmp = t_0;
	} else if (J <= -7e-308) {
		tmp = U;
	} else if (J <= 1.05e-49) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, (U * (0.5 / J))));
	double tmp;
	if (J <= -1e-156) {
		tmp = t_0;
	} else if (J <= -7e-308) {
		tmp = U;
	} else if (J <= 1.05e-49) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * ((J * math.cos((K / 2.0))) * math.hypot(1.0, (U * (0.5 / J))))
	tmp = 0
	if J <= -1e-156:
		tmp = t_0
	elif J <= -7e-308:
		tmp = U
	elif J <= 1.05e-49:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(Float64(J * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U * Float64(0.5 / J)))))
	tmp = 0.0
	if (J <= -1e-156)
		tmp = t_0;
	elseif (J <= -7e-308)
		tmp = U;
	elseif (J <= 1.05e-49)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, (U * (0.5 / J))));
	tmp = 0.0;
	if (J <= -1e-156)
		tmp = t_0;
	elseif (J <= -7e-308)
		tmp = U;
	elseif (J <= 1.05e-49)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1e-156], t$95$0, If[LessEqual[J, -7e-308], U, If[LessEqual[J, 1.05e-49], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.00000000000000004e-156 or 1.0499999999999999e-49 < J

   1. Initial program 85.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. associate-*l* 85.3%

         \[\leadsto \color{blue}{\left(-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. associate-*l* 85.3%

         \[\leadsto \color{blue}{-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]

      3. unpow2 85.3%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]

      4. hypot-1-def 95.7%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]

      5. associate-/r* 95.6%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{\frac{U}{2 \cdot J}}{\cos \left(\frac{K}{2}\right)}}\right)\right) \]

      6. cos-neg 95.6%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(-\frac{K}{2}\right)}}\right)\right) \]

      7. distribute-frac-neg 95.6%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2 \cdot J}}{\cos \color{blue}{\left(\frac{-K}{2}\right)}}\right)\right) \]

      8. associate-/r* 95.7%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{-K}{2}\right)}}\right)\right) \]

      9. associate-/r* 95.6%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{\frac{U}{2 \cdot J}}{\cos \left(\frac{-K}{2}\right)}}\right)\right) \]

      10. distribute-frac-neg 95.6%

          \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2 \cdot J}}{\cos \color{blue}{\left(-\frac{K}{2}\right)}}\right)\right) \]

      11. cos-neg 95.6%

          \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2 \cdot J}}{\color{blue}{\cos \left(\frac{K}{2}\right)}}\right)\right) \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)} \]

   4. Taylor expanded in K around 0 82.0%

      \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot \frac{U}{J}}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 82.0%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{0.5 \cdot U}{J}}\right)\right) \]

      2. *-commutative 82.0%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\color{blue}{U \cdot 0.5}}{J}\right)\right) \]

      3. associate-*r/ 81.9%

         \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]

   6. Simplified 81.9%

      \[\leadsto -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]

   if -1.00000000000000004e-156 < J < -7e-308

   1. Initial program 26.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf 46.1%

      \[\leadsto \color{blue}{U} \]

   if -7e-308 < J < 1.0499999999999999e-49

   1. Initial program 40.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0 41.6%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 41.6%

         \[\leadsto \color{blue}{-U} \]

   4. Simplified 41.6%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1 \cdot 10^{-156}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq -7 \cdot 10^{-308}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \end{array} \]

Alternative 4: 67.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -2.75 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 8.5 \cdot 10^{-21}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.85 \cdot 10^{+149}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (* -2.0 (cos (* K 0.5))))))
   (if (<= J -2.75e-18)
     t_0
     (if (<= J -2e-310)
       U
       (if (<= J 8.5e-21)
         (- U)
         (if (<= J 1.85e+149)
           (* (* -2.0 J) (hypot 1.0 (* U (/ 0.5 J))))
           t_0))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * cos((K * 0.5)));
	double tmp;
	if (J <= -2.75e-18) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if (J <= 8.5e-21) {
		tmp = -U;
	} else if (J <= 1.85e+149) {
		tmp = (-2.0 * J) * hypot(1.0, (U * (0.5 / J)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * Math.cos((K * 0.5)));
	double tmp;
	if (J <= -2.75e-18) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if (J <= 8.5e-21) {
		tmp = -U;
	} else if (J <= 1.85e+149) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (U * (0.5 / J)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = J * (-2.0 * math.cos((K * 0.5)))
	tmp = 0
	if J <= -2.75e-18:
		tmp = t_0
	elif J <= -2e-310:
		tmp = U
	elif J <= 8.5e-21:
		tmp = -U
	elif J <= 1.85e+149:
		tmp = (-2.0 * J) * math.hypot(1.0, (U * (0.5 / J)))
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (J <= -2.75e-18)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = U;
	elseif (J <= 8.5e-21)
		tmp = Float64(-U);
	elseif (J <= 1.85e+149)
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(U * Float64(0.5 / J))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = J * (-2.0 * cos((K * 0.5)));
	tmp = 0.0;
	if (J <= -2.75e-18)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = U;
	elseif (J <= 8.5e-21)
		tmp = -U;
	elseif (J <= 1.85e+149)
		tmp = (-2.0 * J) * hypot(1.0, (U * (0.5 / J)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.75e-18], t$95$0, If[LessEqual[J, -2e-310], U, If[LessEqual[J, 8.5e-21], (-U), If[LessEqual[J, 1.85e+149], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if J < -2.75e-18 or 1.84999999999999989e149 < J

   1. Initial program 96.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around inf 86.7%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 86.7%

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)} \]

      2. *-commutative 86.7%

         \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(0.5 \cdot K\right) \]

      3. *-commutative 86.7%

         \[\leadsto \left(J \cdot -2\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)} \]

      4. associate-*r* 86.7%

         \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)} \]

      5. *-commutative 86.7%

         \[\leadsto J \cdot \left(-2 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \]

   4. Simplified 86.7%

      \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \cos \left(0.5 \cdot K\right)\right)} \]

   if -2.75e-18 < J < -1.999999999999994e-310

   1. Initial program 43.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf 42.2%

      \[\leadsto \color{blue}{U} \]

   if -1.999999999999994e-310 < J < 8.4999999999999993e-21

   1. Initial program 41.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0 40.3%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 40.3%

         \[\leadsto \color{blue}{-U} \]

   4. Simplified 40.3%

      \[\leadsto \color{blue}{-U} \]

   if 8.4999999999999993e-21 < J < 1.84999999999999989e149

   1. Initial program 94.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 23.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\right)} \]

      2. expm1-udef 20.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} - 1} \]

   3. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right) \cdot \left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 23.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right) \cdot \left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 99.6%

         \[\leadsto \color{blue}{\mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right) \cdot \left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}}\right) \cdot \left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \]

      4. *-commutative 99.6%

         \[\leadsto \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(0.5 \cdot K\right)}\right) \cdot \left(J \cdot \left(-2 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(0.5 \cdot K\right)}\right) \cdot \left(J \cdot \left(-2 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]

   6. Taylor expanded in K around 0 78.3%

      \[\leadsto \mathsf{hypot}\left(1, U \cdot \frac{0.5}{\color{blue}{J}}\right) \cdot \left(J \cdot \left(-2 \cdot \cos \left(0.5 \cdot K\right)\right)\right) \]

   7. Taylor expanded in K around 0 67.6%

      \[\leadsto \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right) \cdot \color{blue}{\left(-2 \cdot J\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.75 \cdot 10^{-18}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 8.5 \cdot 10^{-21}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.85 \cdot 10^{+149}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 66.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{-48} \lor \neg \left(J \leq 1.4 \cdot 10^{+22}\right) \land J \leq 2.3 \cdot 10^{+57}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (* -2.0 (cos (* K 0.5))))))
   (if (<= J -7.6e-17)
     t_0
     (if (<= J -2e-310)
       U
       (if (or (<= J 9.2e-48) (and (not (<= J 1.4e+22)) (<= J 2.3e+57)))
         (- U)
         t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * cos((K * 0.5)));
	double tmp;
	if (J <= -7.6e-17) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if ((J <= 9.2e-48) || (!(J <= 1.4e+22) && (J <= 2.3e+57))) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * ((-2.0d0) * cos((k * 0.5d0)))
    if (j <= (-7.6d-17)) then
        tmp = t_0
    else if (j <= (-2d-310)) then
        tmp = u
    else if ((j <= 9.2d-48) .or. (.not. (j <= 1.4d+22)) .and. (j <= 2.3d+57)) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * Math.cos((K * 0.5)));
	double tmp;
	if (J <= -7.6e-17) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if ((J <= 9.2e-48) || (!(J <= 1.4e+22) && (J <= 2.3e+57))) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = J * (-2.0 * math.cos((K * 0.5)))
	tmp = 0
	if J <= -7.6e-17:
		tmp = t_0
	elif J <= -2e-310:
		tmp = U
	elif (J <= 9.2e-48) or (not (J <= 1.4e+22) and (J <= 2.3e+57)):
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (J <= -7.6e-17)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = U;
	elseif ((J <= 9.2e-48) || (!(J <= 1.4e+22) && (J <= 2.3e+57)))
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = J * (-2.0 * cos((K * 0.5)));
	tmp = 0.0;
	if (J <= -7.6e-17)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = U;
	elseif ((J <= 9.2e-48) || (~((J <= 1.4e+22)) && (J <= 2.3e+57)))
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -7.6e-17], t$95$0, If[LessEqual[J, -2e-310], U, If[Or[LessEqual[J, 9.2e-48], And[N[Not[LessEqual[J, 1.4e+22]], $MachinePrecision], LessEqual[J, 2.3e+57]]], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -7.6000000000000002e-17 or 9.2000000000000003e-48 < J < 1.4e22 or 2.2999999999999999e57 < J

   1. Initial program 94.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around inf 79.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 79.9%

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)} \]

      2. *-commutative 79.9%

         \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(0.5 \cdot K\right) \]

      3. *-commutative 79.9%

         \[\leadsto \left(J \cdot -2\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)} \]

      4. associate-*r* 79.9%

         \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)} \]

      5. *-commutative 79.9%

         \[\leadsto J \cdot \left(-2 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \]

   4. Simplified 79.9%

      \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \cos \left(0.5 \cdot K\right)\right)} \]

   if -7.6000000000000002e-17 < J < -1.999999999999994e-310

   1. Initial program 43.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf 42.2%

      \[\leadsto \color{blue}{U} \]

   if -1.999999999999994e-310 < J < 9.2000000000000003e-48 or 1.4e22 < J < 2.2999999999999999e57

   1. Initial program 49.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0 44.8%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 44.8%

         \[\leadsto \color{blue}{-U} \]

   4. Simplified 44.8%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{-48} \lor \neg \left(J \leq 1.4 \cdot 10^{+22}\right) \land J \leq 2.3 \cdot 10^{+57}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 48.6% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -1.05 \cdot 10^{+68}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.36 \cdot 10^{+159}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -1.05e+68)
   (* -2.0 J)
   (if (<= J -2e-310) U (if (<= J 1.36e+159) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.05e+68) {
		tmp = -2.0 * J;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if (J <= 1.36e+159) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-1.05d+68)) then
        tmp = (-2.0d0) * j
    else if (j <= (-2d-310)) then
        tmp = u
    else if (j <= 1.36d+159) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.05e+68) {
		tmp = -2.0 * J;
	} else if (J <= -2e-310) {
		tmp = U;
	} else if (J <= 1.36e+159) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -1.05e+68:
		tmp = -2.0 * J
	elif J <= -2e-310:
		tmp = U
	elif J <= 1.36e+159:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -1.05e+68)
		tmp = Float64(-2.0 * J);
	elseif (J <= -2e-310)
		tmp = U;
	elseif (J <= 1.36e+159)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -1.05e+68)
		tmp = -2.0 * J;
	elseif (J <= -2e-310)
		tmp = U;
	elseif (J <= 1.36e+159)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -1.05e+68], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -2e-310], U, If[LessEqual[J, 1.36e+159], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.05e68 or 1.36e159 < J

   1. Initial program 99.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around inf 93.7%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 93.7%

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)} \]

      2. *-commutative 93.7%

         \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(0.5 \cdot K\right) \]

      3. *-commutative 93.7%

         \[\leadsto \left(J \cdot -2\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)} \]

      4. associate-*r* 93.7%

         \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)} \]

      5. *-commutative 93.7%

         \[\leadsto J \cdot \left(-2 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \]

   4. Simplified 93.7%

      \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \cos \left(0.5 \cdot K\right)\right)} \]

   5. Taylor expanded in K around 0 51.7%

      \[\leadsto \color{blue}{-2 \cdot J} \]

   if -1.05e68 < J < -1.999999999999994e-310

   1. Initial program 52.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf 40.2%

      \[\leadsto \color{blue}{U} \]

   if -1.999999999999994e-310 < J < 1.36e159

   1. Initial program 62.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0 37.0%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 37.0%

         \[\leadsto \color{blue}{-U} \]

   4. Simplified 37.0%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.05 \cdot 10^{+68}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.36 \cdot 10^{+159}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 7: 38.2% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J -2e-310) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -2e-310) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2d-310)) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -2e-310) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -2e-310:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -2e-310)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -2e-310)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -2e-310], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < -1.999999999999994e-310

   1. Initial program 67.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf 29.7%

      \[\leadsto \color{blue}{U} \]

   if -1.999999999999994e-310 < J

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0 27.5%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 27.5%

         \[\leadsto \color{blue}{-U} \]

   4. Simplified 27.5%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8: 26.8% accurate, 420.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ U \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 U)

C

U = abs(U);
double code(double J, double K, double U) {
	return U;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	return U;
}

Python

U = abs(U)
def code(J, K, U):
	return U

Julia

U = abs(U)
function code(J, K, U)
	return U
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	tmp = U;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := U

Derivation

1. Initial program 70.3%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Taylor expanded in U around -inf 28.1%

   \[\leadsto \color{blue}{U} \]

3. Final simplification 28.1%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023299 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))